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This document provides a list of commonly proposed methods for electing an entire legislature.  Because their are so many variations and hybrid methods, these definitions are looser than those for the single-winner methods.
 

Legislature Election Methods
Name:  Single Winner with Districting
Definition:
This is a very general class of method.  The represented area is divided into arbitrary geographic regions, or districts.  One candidate is elected from each district using a single winner method.  In English speaking countries, plurality is often combined with districting.

Name:  STV
Definition:
Each voter fills out a rank ballot.  A process is carried out where if any candidate's number of first place votes is greater than a certain number of votes called a quota, then that candidate is elected.  Whenever candidates are elected, they are eliminated from consideration on all ballots, such that more first-place votes are revealed.  That is, first-place refers to first-place among non-eliminated candidates.

If no candidate meets quota, the candidate with the lowest number of first-place votes is eliminated from consideration without getting a seat.  The process continues until all seats are filled.  If all ranked candidates on a particular vote have been eliminated, this vote is no longer considered for the quota formula.

Different possibilities for quota formulas have been suggested

Let Q be the quota
Let N be the number of seats available
Let T be the total number of votes
floor(x) is defined to be the largest integer less than x

Hare:

Q=floor(T/N)

Q=floor(T/(N+1)+1)

Q=floor(T/(N+2)+1)

Name:  Limited Voting
Single Non-transferable vote (SNTV)
Definition:
Each voter votes for one candidate.  The winning candidates are simply those with the greatest number of votes, in order, until all seats are filled.  This is viewed as a method which encourages proportional representation because it tends to be an optimal strategy for a party to run a number of candidates proportional to its percentage of the vote.  Then, if parties are correct in their guesses, the results will be largely proportional.

Typically large areas are divided into separate multi-candidate districts.

Name:  Cumulative Voting
Definition:
Each voter has a specific number of votes to divide between the candidates as he or she wishes.  The winning candidates are simply those with the greatest number of votes, in order, until all seats are filled.  This is viewed as a method which encourages proportional representation because it tends to be an optimal strategy for a party to run a number of candidates proportional to its percentage of the vote.  Then, if parties are correct in their guesses, the results will be largely proportional.

Typically large areas are divided into separate multi-candidate districts.

Name:  Closed Party List
Definition:
Each party prepares an ordered list of candidates for the legislature.  Voters then vote for a party's list.  A party's candidates are elected, in order from the list.  Sometimes a rule is imposed requiring a party to receive a minimum number of votes to get any representation.

Let N be the number of seats available
Let A be the number of seats allocated to the party
Let V be the number of votes for the party
Let T be the total number of votes
floor(x) is defined to be the largest integer less than x
x mod y is the remainder when x is divided by y

Each party is given a number of seats equal to floor(V/T*N).  This will likely leave some seats unallocated.  Different methods have been suggested for allocating the remaining seats.

Hare/Largest Remainder method
Order the party's from greatest to least based on V mod T*n.  Each party in order is allocated one seat until no more seats remain.

Sainte-Lague
Find the party whose score for V/(A+.5) is the largest.  Allocate one more seat to this party and repeat the process.

D'Hondt
Find the party whose score for V/(A+1) is the largest.  Allocate one more seat to this party and repeat the process.

Name:  Open Party List
Definition:
Each party submits an un-ordered list of candidates.  Each voter votes for one candidate from some party's list.  Each party wins a number of seats based on its proportion of the vote, as in Closed Party List.  The candidates who are elected for a given party are those who receive the most votes.
This method can be viewed as a hybrid between Closed Party List and Cumulative voting.

Name:  Mixed Member Proportional (MMP)
Definition:
Voters vote for some form of open or closed list election and some form of single winner with districting at the same time.  The legislature is divided into two sections, with the outcome of the seats in the legislature being determined by one of the two concurrent elections.

Name:  Additional Member Proportional (AMP)
Definition:
Voters vote for some form of open or closed list election and some form of single winner with districting at the same time.  The support a party wins in the districted election is considered with regard to determining proportionality in the list election.  That is, the list seats are used to make the over-all results as proportional as possible.

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